Why does Nick agree to arrange a meeting between Daisy and Gatsby in The Great Gatsby?
Gatsby has been holding his extravagant parties in the hopes of attracting Daisy to one of them so that he will have a chance to meet her again after all these years of separation. Then when Nick happens to rent a cottage next door to Gatsby's mansion and Gatsby learns that Nick is related to Daisy, it is natural for Gatsby to want to cultivate Nick's friendship and try in a roundabout way to get Nick to set up a rendezvous in Nick's cottage. This puts Nick in an awkward position, since he must realize that Gatsby is interested in starting a secret affair with Daisy, although Nick may not know that Gatsby has much more than that in mind. Gatsby wants to steal Daisy from her husband Tom Buchanan. Gatsby is both bold and timid, poised and awkward, brazen and shy. He does not ask Nick directly to let him use Nick's cottage for a rendezvous. That would make Nick look like a pandar, a go-between, and Nick is very sensitive about moral choices. At the end of Chapter IV, Jordan Baker, with whom Nick is becoming intimate, finally puts the question to Nick at Gatsby's request.
"He wants to know," continued Jordan, "if you'll invite Daisy to your house some afternoon and then let him come over."
"Did I have to know all this [about Gatsby's and Daisy's past relationship] before he could ask such a little thing."
"He's afraid, he's waited so long. He thought you might be offended."
Nick should be offended. He is being used. But he apparently agrees to the request, since he will be with Gatsby and Daisy as a chaperone. It turns out in Chapter V that Nick leaves them alone in Gatsby's mansion after Gatsby has shown them around the place--but Nick as narrator of the entire story makes it clear that Daisy wants him to leave them alone. Chapter V ends with the following paragraph, showing that Gatsby has finally gotten exactly what he wanted and that Daisy is dismissing Nick with a gesture to show him that it is perfectly all right for him to leave them for the romantic interlude which is sure to follow--although any such graphic details would not be described in a novel published in the 1920s.
They had forgotten me, but Daisy glanced up and held out her hand. Gatsby didn't know me now at all. I looked once more at them and they looked back at me, remotely possessed by intense life. Then I went out of the room and down the marble steps into the rain, leaving them there together.
When Daisy holds out her hand, it is both a farewell gesture and a sign that it is perfectly all right with her for Nick to leave them alone. In fact, the handshake might also be a gesture of thanks.
By the time Nick agrees to help Gatsby meet Daisy, Nick has unintentionally learned quite a bit about the two of them as separate individuals. Nick has become painfully aware of the bad marriage between his cousin Daisy and Tom. Nick has observed the friendship between Daisy and Jordan Baker, and shared time at one of Gatsby's parties with Jordan.
All the pieces come together when Gatsby realizes that Nick is related to Daisy. After confirming that relationship through conversation with Jordan, Gatsby convinces Jordan that she must ask Nick to invite Daisy to visit so that Nick's next-door neighbor can "happen" to drop by while she is visiting.
He wants her to see his house...And your house is right next door...I think he half expected her to wander into one of his parties, some night...you should have heard the elaborate way he worked up to it...And Daisy ought to have something in her life.
Nick agrees to host the meeting because he doesn't like to judge people and because he does want to help his cousin and his neighbor.